Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by $$x^4-y^4-z^4+w^4 = 0$$ and consider the involution $$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$ The surface $X$ can be seen as a narural elliptic fibration over $\mathbb{P}^{1}$ as explained here

construct the elliptic fibration of elliptic k3 surface

The quotient $X/i$ inherits a fibration structure over $\mathbb{P}^{1}$ whose generic fiber is a smooth rational curve and with six special fibers which are union of two $\mathbb{P}^{1}$'s intersecting in a point.

Can one give an explicit description of this quotient?

Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by $$x^4-y^4-z^4+w^4 = 0$$ and consider the involution $$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$ The surface $X$ can be seen as a narural elliptic fibration over $\mathbb{P}^{1}$ as explained here

construct the elliptic fibration of elliptic k3 surface

The quotient $X/i$ inherits a fibration structure over $\mathbb{P}^{1}$ whose generic fiber is a smooth rational curve and with six special fibers which are union of two $\mathbb{P}^{1}$'s intersecting in a point.

Can one give an explicit description of this quotient?